Difference between revisions of "Glossary"
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| + | === Basic Morita/stable equivalence === | ||
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| + | Morita/stable equivalence of blocks induced by a bimodule which has endopermutation source. | ||
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| + | === CFSG === | ||
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| + | The classification of finite simple groups. | ||
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| + | === Fusion system === | ||
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| + | Fusion system on a finite <math>p</math>-group. See [[References|[Cr11] or [AKO11]]]. The fusion system given by a finite group <math>G</math> on a Sylow <math>p</math>-subgroup <math>P</math> is written <math>\mathcal{F}_P(G)</math>. The Fusion system for a block <math>B</math> (sometimes calle the Brauer category) is defined with respect to a maximal subpair <math>(D,b_D)</math>, and is written <math>\mathcal{F}_{(D,b_D)}(G,B)</math>. | ||
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| + | === Height of an irreducible character === | ||
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| + | An irreducible character <math>\chi</math> in a block <math>B</math> of <math>\mathcal{O} G</math> with defect group <math>D</math> has height <math>h</math> if <math>\chi(1)_p=p^h[G:D]_p</math>. | ||
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| + | === Index p covering blocks === | ||
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| + | Fix a Morita equivalence class <math>M</math>. This lists Morita equivalence classes containing a block <math>B</math> of <math>kG</math> for some finite group <math>G</math> such that <math>B</math> covers a block <math>b</math> in <math>M</math> of <math>kN</math> for some normal subgroup <math>N</math> of <math>G</math> of index <math>p</math>. | ||
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| + | === Isotypy === | ||
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| + | A family of perfect isometries resulting from the existence of a splendid derived equivalence. Introduced by Broué. See [[References#L|[Li18d,9.5]]]. | ||
=== # lifts / <math>\mathcal{O}</math> === | === # lifts / <math>\mathcal{O}</math> === | ||
The number of <math>\mathcal{O}</math>-Morita equivalence classes of blocks reducing to a representative of the given <math>k</math>-class. | The number of <math>\mathcal{O}</math>-Morita equivalence classes of blocks reducing to a representative of the given <math>k</math>-class. | ||
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| + | === MNA(r,s) === | ||
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| + | A class of minimal nonabelian <math>2</math>-groups, that is nonabelian <math>2</math>-groups such that every proper subgroup is abelian. For <math>r \geq s \geq 1</math> | ||
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| + | \[ MNA(r,s) = \langle x,y|x^{2^r}=y^{2^s}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle . \] | ||
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| + | === p'-index covered blocks === | ||
| + | |||
| + | Fix a Morita equivalence class <math>M</math>. This lists Morita equivalence classes containing a block <math>b</math> of <math>kN</math> for some finite group <math>N</math> such that <math>b</math> is covered by a block <math>B</math> in <math>M</math> of <math>kG</math> for some finite group <math>G</math> containing <math>N</math> as a normal subgroup of prime index different to <math>p</math>. | ||
| + | |||
| + | === p'-index covering blocks === | ||
| + | |||
| + | Fix a Morita equivalence class <math>M</math>. This lists Morita equivalence classes containing a block <math>B</math> of <math>kG</math> for some finite group <math>G</math> such that <math>B</math> covers a block <math>b</math> in <math>M</math> of <math>kN</math> for some normal subgroup <math>N</math> of <math>G</math> of prime index different to <math>p</math>. | ||
=== Picard group === | === Picard group === | ||
Let <math>R</math> be a commutative ring and <math>A</math> an <math>R</math>-algebra. The Picard group <math>{\rm Pic}_R(A)</math> has elements isomorphism classes of <math>A</math>-<math>A</math>-bimodules inducing a Morita equivalence, with multiplication given by taking tensor products over <math>A</math>. | Let <math>R</math> be a commutative ring and <math>A</math> an <math>R</math>-algebra. The Picard group <math>{\rm Pic}_R(A)</math> has elements isomorphism classes of <math>A</math>-<math>A</math>-bimodules inducing a Morita equivalence, with multiplication given by taking tensor products over <math>A</math>. | ||
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| + | === Possible Brauer tree (for a given cyclic defect group) === | ||
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| + | Fix a cyclic group <math>P</math> of order <math>p^n</math>. A block with defect group <math>P</math> has inertial index <math>e</math> a divisor of <math>p-1</math>. The number of irreducible characters in the block is <math>e+\frac{|P|-1}{e}</math>. The exceptional vertex has multiplicity <math>\frac{|P|-1}{e}</math>. | ||
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| + | The possibile Brauer trees (for <math>P</math> and <math>e</math> a divisor of <math>p-1</math>) are the Brauer trees whose vertex multiplicities add to <math>e+\frac{|P|-1}{e}</math> where the exceptional vertex multiplicity is <math>\frac{|P|-1}{e}</math> and non-exceptional vertices are regarded as having multiplicity <math>1</math>. | ||
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| + | === Resistant p-group === | ||
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| + | A p-group <math>P</math> is resistant if whenever <math>\mathcal{F}</math> is a saturated fusion system on <math>P</math>, we have <math>\mathcal{F}=N_{\mathcal{F}}(P)</math>, or equivalently <math>\mathcal{F}=\mathcal{F}_P(G)</math> for some finite group <math>G</math> with <math>P</math> as a normal Sylow p-subgroup. Resistant p-groups were introduced in [[References#S|[St02]]] in terms of fusion systems for groups, and for arbitrary saturated fusion systems in [[References#S|[St06]]]. | ||
| + | |||
| + | === Source algebra === | ||
| + | |||
| + | Introduced by Puig, these are subalgebras of a block <math>B</math> of a finite group (and more generally for a <math>G</math>-algebra) not only Morita equivalent to <math>B</math> but also determining the fusion system. See [[References#L|[Li18c,5.6.12]]]. | ||
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| + | === Splendid equivalence === | ||
| + | |||
| + | May apply to Morita equivalences, stable equivalences and derived equivalences. See 9.7 an 9.8 of [[References|[Li18d]]]. It means roughly equivalences given by (complexes of) trivial source bimodules. | ||
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| + | === Trivial intersection subgroup === | ||
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| + | A subgroup <math>H \leq G</math> such that <math>\forall g \in G \setminus N_G(H)</math> we have <math>H^g\cap H=1</math>. | ||
Latest revision as of 22:02, 18 November 2020
Contents
- 1 Basic Morita/stable equivalence
- 2 CFSG
- 3 Fusion system
- 4 Height of an irreducible character
- 5 Index p covering blocks
- 6 Isotypy
- 7 # lifts / [math]\mathcal{O}[/math]
- 8 MNA(r,s)
- 9 p'-index covered blocks
- 10 p'-index covering blocks
- 11 Picard group
- 12 Possible Brauer tree (for a given cyclic defect group)
- 13 Resistant p-group
- 14 Source algebra
- 15 Splendid equivalence
- 16 Trivial intersection subgroup
Basic Morita/stable equivalence
Morita/stable equivalence of blocks induced by a bimodule which has endopermutation source.
CFSG
The classification of finite simple groups.
Fusion system
Fusion system on a finite [math]p[/math]-group. See [Cr11] or [AKO11]. The fusion system given by a finite group [math]G[/math] on a Sylow [math]p[/math]-subgroup [math]P[/math] is written [math]\mathcal{F}_P(G)[/math]. The Fusion system for a block [math]B[/math] (sometimes calle the Brauer category) is defined with respect to a maximal subpair [math](D,b_D)[/math], and is written [math]\mathcal{F}_{(D,b_D)}(G,B)[/math].
Height of an irreducible character
An irreducible character [math]\chi[/math] in a block [math]B[/math] of [math]\mathcal{O} G[/math] with defect group [math]D[/math] has height [math]h[/math] if [math]\chi(1)_p=p^h[G:D]_p[/math].
Index p covering blocks
Fix a Morita equivalence class [math]M[/math]. This lists Morita equivalence classes containing a block [math]B[/math] of [math]kG[/math] for some finite group [math]G[/math] such that [math]B[/math] covers a block [math]b[/math] in [math]M[/math] of [math]kN[/math] for some normal subgroup [math]N[/math] of [math]G[/math] of index [math]p[/math].
Isotypy
A family of perfect isometries resulting from the existence of a splendid derived equivalence. Introduced by Broué. See [Li18d,9.5].
# lifts / [math]\mathcal{O}[/math]
The number of [math]\mathcal{O}[/math]-Morita equivalence classes of blocks reducing to a representative of the given [math]k[/math]-class.
MNA(r,s)
A class of minimal nonabelian [math]2[/math]-groups, that is nonabelian [math]2[/math]-groups such that every proper subgroup is abelian. For [math]r \geq s \geq 1[/math]
\[ MNA(r,s) = \langle x,y|x^{2^r}=y^{2^s}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle . \]
p'-index covered blocks
Fix a Morita equivalence class [math]M[/math]. This lists Morita equivalence classes containing a block [math]b[/math] of [math]kN[/math] for some finite group [math]N[/math] such that [math]b[/math] is covered by a block [math]B[/math] in [math]M[/math] of [math]kG[/math] for some finite group [math]G[/math] containing [math]N[/math] as a normal subgroup of prime index different to [math]p[/math].
p'-index covering blocks
Fix a Morita equivalence class [math]M[/math]. This lists Morita equivalence classes containing a block [math]B[/math] of [math]kG[/math] for some finite group [math]G[/math] such that [math]B[/math] covers a block [math]b[/math] in [math]M[/math] of [math]kN[/math] for some normal subgroup [math]N[/math] of [math]G[/math] of prime index different to [math]p[/math].
Picard group
Let [math]R[/math] be a commutative ring and [math]A[/math] an [math]R[/math]-algebra. The Picard group [math]{\rm Pic}_R(A)[/math] has elements isomorphism classes of [math]A[/math]-[math]A[/math]-bimodules inducing a Morita equivalence, with multiplication given by taking tensor products over [math]A[/math].
Possible Brauer tree (for a given cyclic defect group)
Fix a cyclic group [math]P[/math] of order [math]p^n[/math]. A block with defect group [math]P[/math] has inertial index [math]e[/math] a divisor of [math]p-1[/math]. The number of irreducible characters in the block is [math]e+\frac{|P|-1}{e}[/math]. The exceptional vertex has multiplicity [math]\frac{|P|-1}{e}[/math].
The possibile Brauer trees (for [math]P[/math] and [math]e[/math] a divisor of [math]p-1[/math]) are the Brauer trees whose vertex multiplicities add to [math]e+\frac{|P|-1}{e}[/math] where the exceptional vertex multiplicity is [math]\frac{|P|-1}{e}[/math] and non-exceptional vertices are regarded as having multiplicity [math]1[/math].
Resistant p-group
A p-group [math]P[/math] is resistant if whenever [math]\mathcal{F}[/math] is a saturated fusion system on [math]P[/math], we have [math]\mathcal{F}=N_{\mathcal{F}}(P)[/math], or equivalently [math]\mathcal{F}=\mathcal{F}_P(G)[/math] for some finite group [math]G[/math] with [math]P[/math] as a normal Sylow p-subgroup. Resistant p-groups were introduced in [St02] in terms of fusion systems for groups, and for arbitrary saturated fusion systems in [St06].
Source algebra
Introduced by Puig, these are subalgebras of a block [math]B[/math] of a finite group (and more generally for a [math]G[/math]-algebra) not only Morita equivalent to [math]B[/math] but also determining the fusion system. See [Li18c,5.6.12].
Splendid equivalence
May apply to Morita equivalences, stable equivalences and derived equivalences. See 9.7 an 9.8 of [Li18d]. It means roughly equivalences given by (complexes of) trivial source bimodules.
Trivial intersection subgroup
A subgroup [math]H \leq G[/math] such that [math]\forall g \in G \setminus N_G(H)[/math] we have [math]H^g\cap H=1[/math].
